Lattice path matroids and quotients

TL;DR

The paper develops a complete combinatorial framework for quotients among lattice path matroids (LPMs) by a diagram-based criterion, enabling the construction of a graded quotient poset $\mathcal{P}_n$ whose rank distribution is governed by Narayana numbers. It then studies lattice path flag matroids (LPFMs) and shows they correspond to points in the nonnegative flag variety $\mathcal{F}\ell_n^{\ge 0}$, with LPFM flag polytopes realizing Bruhat-interval polytopes under the moment map. A key contribution is proving a conjecture of Mcalmon, Oh, and Xiang for LPMs: quotients are characterized by CCW-arrow decompositions, confirming realizability and providing a tight combinatorial description. The results bridge lattice path combinatorics, matroid quotients, and geometric aspects of the nonnegative Grassmannian and flag varieties, revealing new structural and enumerative connections and guiding directions for LPFM diagrammatic theory and Higgs-lift questions.

Abstract

We characterize the quotients among lattice path matroids (LPMs) in terms of their diagrams. This characterization allows us to show that ordering LPMs by quotients yields a graded poset, whose rank polynomial has the Narayana numbers as coefficients. Furthermore, we study full lattice path flag matroids and show that -- contrary to arbitrary positroid flag matroids -- they correspond to points in the nonnegative flag variety. At the basis of this result lies an identification of certain intervals of the strong Bruhat order with lattice path flag matroids. A recent conjecture of Mcalmon, Oh, and Xiang states a characterization of quotients of positroids. We use our results to prove this conjecture in the case of LPMs.

Figures (12)

• Figure 1: A basis in the diagram representing the LPM $M[1246,3568]$.
• Figure 2: The lattice of bases of $M[1246,3568]$.
• Figure 3: A diagram representing $M[135,246]=U_{1,2}\oplus U_{1,2}\oplus U_{1,2}$.
• Figure 4: An illustration of Lemmas \ref{['lem:quotient_def']} and \ref{['lem:Bp']}: $B_p$ are the North steps in $B$ that can be turned into East steps, such that if $p$ is turned into a North step, then the resulting path is valid. They are precisely the North steps between $b_s$ and $b_t$.
• Figure 5: An LPM with a bad pair $(\ell_i, u_j)$ . The gray basis $B$ has $(B\setminus\{b_r\})_p\not\subseteq B_p$ for all $r$. Exactly those $\ell\in L$ on the dotted path yield good pairs with $u_j$.

Theorems & Definitions (57)

• Example 1
• Definition 2
• Remark 3
• Definition 4
• Lemma 5
• Definition 6: BGW
• Example 7
• Definition 8
• Lemma 10
• proof